Perpendicular Lines & Planes, Inequality Postulates & Theorems, Triangle Inequality (Quiz #4)

A plane is undefined, but we can describe it as a surface such that if any two points in the surface are joined by a line, then the line lies entirely in the surface

If points, lines, segments lie in the same plane, we call them coplanar. If they do not lie in the same plane, we call them noncoplanar.

Postulate: If a line passes through two points that lie in a plane, then the line lies entirely in the plane

Postulate: If a line intersects a plane not containing it, then they intersect at exactly one point called the foot of the line

Postulate: If two planes intersect, then their intersection is exactly one line

Definition: A line is oblique to a plane if it intersects the plane at exactly one point and is not perpendicular to the plane

Postulate: Three non-collinear points determine a plane

Theorem: A line and a point not on the line determine a plane

Theorem: Two intersecting lines determine a plane

Definition: A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through its foot

Theorem: If a line is perpendicular to at least two distinct lines that lie in a plane and pass through the foot, then it is perpendicular to the plane

Postulates:

• Given a plane and a point on the plane, there is exactly one line passing through the given point that is perpendicular to the plane

• Given a plane and a point not on the plane, there is exactly one line passing through the given point that is perpendicular to the plane

Law of Trichotomy:

For real numbers, exactly one of the following is true:

A<b, a=b, a>b

A whole is greater than any of its parts

a) if point X is between A and B, then AB > AX and AB > BX

b) if point X is in the interior of <ABC, then m<ABC > m<ABX and m<ABX > m<XBC

Transitive property for inequalities:

If a>b and b>c, then a>c

Substitution Property for Inequalities:

A quantity may be substituted for its equal in any inequality

If a>b and b=c, then a>c

If a<b and b=c, then a<c

Addition property for inequalities

• if equal quantities are added to unequal quantities, then their sums are unequal in the same order

• if unequal quantities are added to quantities that are unequal in the same order, then their sums are unequal in the same order

Subtraction property for inequalities

• If equal quantities are subtracted from equal quantities, then their differences are unequal in the same order

• if unequal quantities are subtracted from equal quantities, then their differences are unequal in the opposite order

Multiplication and division properties for inequalities

• doubles of unequal quantities are unequal in the same order

• halves of unequal quantities are unequal in the same order

• if two quantities are unequal and they are multiplied or divided by a positive number, the resulting products are unequal in the same order

• if two quantities are unequal and they are multiplied or divided by a negative number, the resulting products are unequal in the opposite order

Definition: an exterior angle of a triangle is an angle that is adjacent and supplementary to an interior angle of the triangle

Exterior angle inequality theorem:

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle

Side angle inequality theorem:

If two sides of a triangle are not congruent, then the angles opposite them are also not congruent, and furthermore, the largest angle is opposite the longer side

Converse of the side angle inequality theorem:

If two angles of a triangle are not congruent, then the sides opposite them are also not congruent, and furthermore, the longer side is OPP. the larger angle